# Quick Review: Logarithm

###### Formulas

Any numeral is known as a number. Numbers are of various types. Let us discuss the types of numbers.

• Definition: ax = b can be represented in logarithmic form as loga b = x
• log a = x means that 10x = a .
• 10log a = a (The basic logarithmic identity).
• log (ab) = log a + log b, a > 0, b > 0
• log(a/b) = log a -log b, a > 0, b > 0.
• log an = n (log a) (Logarithm of a power).
• logx y = logmy / logmx (Change of base rule).
• logx y = 1 / logy x .
• logx1 = 0 (x ≠ 0, 1).
• The natural numbers 1, 2, 3,.... are respectively the logarithms of 10, 100, 1000, .... to the base 10.
• The logarithm of "0" and negative numbers are not defined.
• logb1= 0 (∵ b0 = 1)
• logb b = 1 (∵ b1 = b)
• y = ln x → ey = x
• x = ey = → ln x = y
• x = ln ex = eln x
• elogb⁡x = x
• logb by = y
###### Laws of Logarithm
• logb⁡MN = logb⁡M + logb⁡N (where b, M and N are positive real numbers and b ≠ 1)
• logb⁡(M/N) = logb⁡M - logb⁡N (where b, M and N are positive real numbers and b ≠ 1)
• logb⁡(M) = c logb⁡M (where b, M and N are positive real numbers and b ≠ 1 and c is any real number)
• logb⁡M = log⁡M/log⁡b = ln⁡M/ln⁡b = logk⁡M/logk⁡b (where b, M and k are positive real numbers and b ≠ 1, k≠1)
• logb⁡a = 1/loga⁡b (where b, and a are positive real numbers and b ≠ 1, a≠1)
• If logb⁡M = logb⁡N, then M = N (where b, M and N are positive real numbers and b ≠ 1)
###### Euler's Number

A mathematical consonant e is the base of the natural logarithm, known as Euler's number. It is also known as Napier's consonant.  